What is a bookmaker

A bookmaker, or bookie, is an organization or a person that takes bets on sporting and other events at agreed upon odds.

Bookmakers make money by offering odds that are different to what they believe are the real probabilities in the underlying event. They set their odds lower than the probability such that they can make a profit no matter what the outcome of the event.

Odds

Odds in the UK are traditionally expressed as 2-1 or 5-2 or 6-4. Decimal (European odds) are much simpler for comparison purposes and are easily calculated form the UK format as follows:
5-2 is 5 divided by 2 = 2.5 and to get the decimal odds you simply add 1 to this to give 3.5
6-4 is 6 divided by 4 which equals 1.25 then add 1 gives the decimal odds of 2.25
2-1 is easy this is 3.0 decimal

Odds less than Probability

A bookmaker may assess a football match as follows .. the probability of the Home team winning is 50%, the probability of the Away team winning is 30% and the probability of a Draw is 20%. The odds that should be offered for these probabilities are:

Probability 50% - the odds should be 100% / 50% = 2.0
Probability 30% - the odds should be 100% / 30% = 3.33
Probability 20% - the odds should be 100% / 20% = 5.0

The odds that the bookie will offer for this event might look something like this:
Home win - instead of 2.0 they'll offer 1.8
Away win - instead of 3.33 they'll offer 3.0
Draw - instead of 5.0 they'll offer 4.5

Oxford and Cambridge Boat Race

Our bookmaker decides to accept betting on an event where there are only two outcomes. He decides to choose the the Oxford and Cambridge boat race.

The bookmaker might open his book by offering odds of 1-2 for Cambridge and 6-4 for Oxford. 

Cambridge are the big favourites and the chances of Cambridge winning = 2/(1+2) = 2/3 = 66.67%.
Oxford's chances of winning are 4 out of every 10 = 40%.

The probabilities add up to 106.67% The excess of 6.67% over 100% is known as the over-round (or vig or juice). In short he has sold odds of 106.67% but the outcome can only ever be 100%. There is a 100% chance that one horse or the other will win. (We are talking simplistically here). The bookmaker, if he can take bets in the proportion of the probabilities, say £66.67 on Cambridge and £40 on Oxford, will pay out £100 whichever wins on £106.67 taken, a percentage profit to him of 6.25%.

In practice, of course, the bookmaker will need to adjust his odds in accordance with supply and demand.  More money for Oxford than the estimated probability indicates will cause him to shorten the odds against Oxford and lengthen those for Cambridge. The same principle works on the stock market when market makers buy and sell shares on their books at different bid and offer prices.